Onn-th commutators with nilpotent or regular values in rings

Then-th commutator for a,b in a ringR is defined inductively as follows: a,b]₁=a,b]=ab−ba anda,b] _n=a,b]₋₁,b]. We characterize the ringsR without non-zero nil right ideals in whicha,b] _nis nilpotent or regular for alla,b∈R. We also examine the case whereR is a semiprime ring with involution in whicht ₁, t₂]_nis nilpotent or regular for all tracest ₁,t₂∈R.